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Applied Arithmetic

Past Questions

Mrs. A. Gilleran ©

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Contents

Arithmetic ............................................................................................................................................... 3

Percentages ............................................................................................................................................. 7

VAT: Value Added Tax ........................................................................................................................... 15

Foreign Currency ................................................................................................................................... 16

Ratio and Proportion............................................................................................................................. 21

Income Tax ............................................................................................................................................ 27

Interest/Compound Interest ................................................................................................................. 38

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Arithmetic

Household Bills

These are charged by the amount of electricity or gas used within a period of time

Number of units used = present meter reading – previous meter reading

Cost of Units used = number of units used x rate per unit

2012- Q2

Fuel consumption in a car is measured in litres per 100 km. Alan’s car travels 1250 km on a tank of 68 litres.

Calculate his car’s fuel consumption in litres per 100km.

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Q9

Consideration is being given to changing the number of points a team gets for a win and also the number of points a team gets for a draw in a soccer league. No points will be awarded for a loss. The table below shows the standing of two teams after six games under the proposed new system.

Team Played Won Drawn Lost Points

(new system)

Team A 6 2 2 2 12

Team B 6 1 5 0 10

(a) Find the number of points which would be awarded for (i) a win and (ii) a draw under this proposed system.

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(b) The current system awards 3 points for a win and 1 point for a draw. Suggest one reason why it might be preferable to change to the system propped in part (a)

2009 – Q1

A supermarket has a special offer on three different brands of packets of soap.

The following table gives details of this offer:

Brand No. of bars per packet Weight of each bar Price of packet

A 3 100g €1.35

B 6 100g €2.40

C 4 125g €2.38

Which brand has the cheapest price per gram?

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2008 – Q1

Three students rent a house for a period of 8 months.

The refuse charges are €16.80 per month.

The electricity bill amounts to €84 every two months.

The television and broadband charges are €324 for the period of the rental.

How much should each of the three students pay monthly for these charges?

2005 – Q2

(ii) The monthly line rental on Peter’s mobile phone amounts to €12.70. During May, the duration of his calls is 1 hr 41 mins and 50 secs. Calls are charged at 0.6 cent per second. Calculate Peter’s total bill for May.

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Percentages

% Profit (Mark-Up).

This is the profit as a percentage of the cost price.

% Profit = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃/𝐿𝐿𝑃𝑃𝐿𝐿𝐿𝐿𝐶𝐶𝑃𝑃𝐿𝐿𝑃𝑃 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

× 100

% Profit(Margin)

This is the profit as a percentage of the selling price.

% Profit = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑆𝑆𝑃𝑃𝑆𝑆𝑆𝑆𝑃𝑃𝑆𝑆𝑆𝑆 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

× 100

Note: When dealing with percentages the denominator (bottom) is always the starting/Original

amount.

2016-Q1

(a) A bus company increases the price of all its tickets by 6%.

Before the increase, the price of a ticket from Cork to Dublin was €17.00.

(i) Find the price of this ticket after this increase.

Six months later, the company reduces the price of this ticket back to €17.00.

(ii) Find the percentage decrease in the price of this ticket. Give your answer correct to

one decimal place.

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2015-Q2

(a) David weighs 88 kg. The average male triathlete of his height weighs 83 kg. If David aims to reach this weight, what percentage decrease is required? Give your answer correct to two decimal places.

(b) Mary’s house was worth €200,000. Mary increase the value of her house by 15% by building a conservatory. She then increased its value by a further 10% by repaving the driveway. Find the total percentage increase in value.

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2013-Q4

The minimum was per hour for different categories of workers is shown in the table. By law the Under 18 minimum wage is set at 70% of the minimum wage for an experienced adult worker.

(a) Verify that this is true for the rates shown in the table on the right.

(b) The government has decided that it is going to reduce all minimum wage rates by 6%. Calculate the new minimum wage for an experienced adult worker, correct to two decimal places, after this reduction.

Category Min. Wage per hour Experienced adult worker €8.65 Aged under 18 €6.06 Over 18 in first year from date of first employment

€6.92

Over 18 in second year of first employment

€7.79

Source: www.citizensinformation.ie

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(c) John is an experienced adult worker. After the reduction he says “If the minimum wage were to be increased by 6% then I would be back earning €8.65 per hour.”. Is John’s statement correct? Explain your answer.

2013 – Q1

Adam got 24 marks from a total of 30 marks in a class test.

What percentage mark did Adam get?

2011- Q2

A computer salesperson is paid an annual salary of €30,000. He is also paid a commission of 4% on sales.

Last year the salesperson earned €38,000.

Calculate the value of sales.

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2010 - Q1

The price of a litre of petrol on the 1st of August was €1.20. The price on the 1st of September was €1.17.

Calculate the percentage decrease over this period.

2009-Q1

In a school library, 28% of the books are classified as fiction and the remainder as non-fiction. There are 3240 non-fiction books in the library.

Find the number of books which are classified as fiction.

2008 -Q2

In 2006, the average costs of running a car for the year were as follows:

Road tax €485, petrol €1440, servicing €650 and insurance €425.

(i) What was the total cost of running the car in 2006?

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In 2007, the petrol went up by 5%, the cost of servicing went up by 15% and the cost of insurance went down by 10%.

(ii) Given that the total running costs increased by 4.6% in 2007, calculate the percentage (%) increase in road tax for 2007, giving your answer correct to one decimal place.

2007-Q1

A snowman has a mass of 12 kg.

It melts at a rate of 0.2% of its mass per minute.

What will the mass of the snowman be after 3 minutes? Give your answer correct to 2 decimal places.

Q2. An auctioneer sells a house for €830,000. The auctioneer’s fee is 1.5% on the first €500,000 and

2.5% on the remainder. Calculate the auctioneer’s fee.

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2006-Q1

The price of a DVD increases from €12.50 to €13.75.

Express this increase as a percentage of the original price.

Q2.

Derek processed 390 passport applications during the month of July. He processed 10% fewer applications during the month of August.

How many applications did he process in August?

2005-Q2 There are25000 fish in a fish farm.

(i) The number of fish in the farm increases by 40% each year.

How many fish will be in the farm at the end of 3 years?

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2004-Q2

The price of a playstation game is €59.99.

In a sale the price of this playstation game is reduced to €49.99.

What is the percentage reduction on the original price of the game in the sale?

Give your answer correct to the nearest whole number.

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VAT: Value Added Tax - This is a tax on goods and services. The rate of VAT is given in the form of a

percentage, e.g. 15% - Sometimes prices are given inclusive of VAT. This means that VAT has already been

added on. - Sometimes prices are given exclusive of VAT. The VAT has not been added on.

Percentages Greater than 100%

A meal in a restaurant cost Jerry €136.20. The price included VAT of 13.5%. What was the price of the meal before VAT was included.

Cost Price + VAT = €136.20

100% + 13.5% = €136.20

113.5% = €136.20

1%=€136.20113.5

=€1.20

100% = €1.20 x 100 = €120

2014-Q5

A meal in a restaurant costs Jerry €136.20. The price included VAT at 13.5%. Jerry wished to know the price of the meal before the VAT was included. He calculated 13.5% of €136.20 and subtracted it from the cost of the meal.

(i) Explain why Jerry will not get the correct answer using this method.

(ii) From July 1, 2011, the VAT rate on food, in restaurants, was reduced to 9%. How much would Jerry have paid for the meal after this date if the VAT reduction was correctly applied?

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Foreign Currency

“2FM” – To Foreign Multiply i.e. when you are converting to a foreign currency multiply the rates, when you are converting back to the home currency divide.

2012 - Q2 (c)

An electronics company imports tablet computers from China at a cost of 696 Yuan ¥ per tablet.

(i) Find the cost of each tablet, in euro, if €1 = 8.7¥

The company must also pay a shipping cost on each tablet imported.

By selling a tablet at €105.40, the company can make a profit of 24%.

(ii) Find the shipping cost per tablet.

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The company imports 1000 tablets from China. It sells 600 of them at €105.40 each (i.e. at a profit of

24%) and the remained at a profit of 15%.

(iii) Find the overall profit, in euro, made by the company.

2012-Q3

The value of one euro against other currencies on a particular day is shown in the table below.

Currency Rate (€) US Dollar 1.4045 Pound Sterling 0.87315 Lithuanian Litas 3.4528 Latvian Lats 0.7093 Polish Zloty 4.0440

(a) Mary was going to America for a few months. She changed €1200 into US Dollars using the exchange rate in the table.

(i) How many dollars should she receive at this exchange rate?

(ii) The bank charged 3% commission on the transaction. How many dollars did she receive?

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(b) On returning to Ireland Mary had €3060. She changed that amount into euro. The bank again charged her 3% commission on the transaction. She received €2047. Find the exchange rate on that, correct to two decimal places.

(c) David changed a certain amount of sterling into euro at the exchange rage in the table above. A few days later he again changed the same amount of sterling into euro at a different exchange rate. He received fewer euro this time. No commission was charged on these transactions. Write down one possible value for the exchange rate for the second transaction.

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2011- Q1 (c) (ii)

Úna and Conor were travelling to South Africa. They bought 5760 rand in the bank.

The bank charged them €630, which included a 5% service charge.

What was the value of the euro in rand (the exchange rate) on that day?

2006 - Q2

(i) A merchant buys tea for €3.29 per kg and then sells it a profit of 60% of the cost price to a customer in England. The exchange rate is £1 (sterling) = €1.46 Calculate the selling price of the tea in £ sterling, correct to two decimal places.

(ii) The exchange rate changes to £1 (sterling) = €1.50. The selling price, in sterling, remains the same. Calculate the merchant’s percentage profit in this case, correct to the nearest whole number.

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2005- Q1

A tourist paid $4620 to a travel agent for a holiday in Ireland, where €1 = $1.32. The cost to the travel agent of organising the holiday was €2985. Calculate, in euro, the profit made by the travel agent.

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Ratio and Proportion

To split amounts into ratios do the following:

• Add up the ratios to get the total numbers of parts

• Divide the total amount to be split by the sum of the ratios to get the amount per part

• Multiply each ratio by the amount per part.

Inverse Proportion:

Multiplying the quantities will always give the same answer. If one number is increased, the other

must be reduced.

Q2.

Eight workers can build a cabin in 60 hours.

How many workers are needed if the cabin is to be built in 32 hours.

2016- Q2

(a)

(i) Write 20 cent as a fraction of €20. Give your answer in its simplest form.

(ii) Write 0.5 cm as a fraction of 2m. Give your answer in its simplest form.

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(b) A shop sells two brands of orange juice. Brand A and Brand B, as shown. (i) Find which brand, A or B, is cheaper per litre. Show all

of your working out.

Samantha needs to buy at least 5 litres of orange juice.

(ii) Find the lowest price that she could pay to do this, by buying Brand A, Brand B, or a combination of both. Justify your answer fully.

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2012-Q4

A soccer team has three strikers John, Paul and Michael. The number of minutes each had played by the end of a particular season is shown on the table. The team divided a bonus of €150,000 between its strikers in proportion to the time each had played.

(a) Calculate the amount each player received.

(b) At the end of the following season a larger total bonus was paid. At that time, John said: “The bonus should be paid according to the number of goals scored by the striker. Paul scored 50% more goals than Michael. I scored as many as both of them together. I would get €140,000 if the team used this method.”

(i) Calculate the total bonus on offer that season.

Name Minutes Played John 2250 Paul 2600 Michael 150

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(ii) How much each would Paul and Michael get under John’s new system?

Q2. The lengths of two pieces of timber are in a ratio of 5:2. The larger piece measures 250mm.

Find the length of the shorter piece.

2011-Q1

Peter and Anna share a lotto prize in the ratio 3 12 to 2 1

2.

Peter’s share is €35,000.

What is the total prize fund?

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2004-Q1

The area of a house covers 205 𝑚𝑚2.

The area of the site for the house covers 1025 𝑚𝑚2.

What is the ratio of the area of the house to the area of the site? Give your answer in the form 1: n where 𝑛𝑛 ∈ ℕ

Q2.

Two brands of blackcurrant squash drinks contain concentrated juice and sugar.

In brand A, the ratio of concentrated juice to sugar is 19:1.

In brand B, the ratio of concentrated juice to sugar is 9:1.

(i) What is the volume of concentrated juice in 500 ml of brand A?

(ii) What is the volume of sugar in 300 ml of brand B?

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500 ml of brand A is mixed with 300 ml of brand B.

(iii) What is the ratio of the concentrated juice to the sugar in the mixture?

2009-Q2

Three business partners, Aideen, Brian and Caroline, invest €30000, €40,000 and €70,000 respectively. At the end of each year, 22.5% of the profit made is placed in reserve and the remainder is divided among the partners in proportion to their investments.

(i) Given that in 2007, the profit amounted to €12,880, calculate the amount placed in reserve.

(ii) In 2008, Caroline’s portion of the profit was €9331. Calculate how much Aideen and

Brian each received in 2008.

(iii) Calculate the amount placed in reserve in 2008.

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Income Tax

Income Tax

Box Method!!!

Gross Tax – Tax Credits = tax payable

Gross income is the amount of money earned before any deductions are made

Net Income (take – home – pay) is the amount of money left after all deductions have been taken away.

Net Income = Gross Income – Tax Paid

USC = Universal Social Charge (statutory deduction)

Tax credits: Tax you do not have to pay

Question 3

Eleanor has a gross income of €38,500 for the year.

She has an annual tax credit of €3,300.

The standard rate cut-off point is €33,800.

The standard rate of income tax is 20% and the higher rate is 40%.

(a) Find Eleanor’s net income tax for the year (i.e. after tax is paid).

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Eleanor receives a pay rise. As a result, her net income for the year is €34,780.

(b) Find Eleanor’s new gross income for the year.

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2014-Q7

Last year Elena had a gross income of €36, 960. She had to pay Universal Social Charge (USC) and income tax on her gross income. The rates and bands of USC are as follows.

Income Band Rate of USC

Up to €10,036 2%

Between €10,036 and €16,016 4%

Above €16,016 7%

(i) Find the amount of USC that was deducted from Elena’s gross income last year.

(ii) The standard rate of income tax was 20% and the higher rate was 41%.

The standard rate cut-off point was €32,800. Elena paid a total of €4965.60 income tax last year. Find Elena’s tax credits for the year.

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(iii) Find Elena’s total deduction (USC and income tax) as a percentage of her gross income. Give your answer correct to the nearest percent.

2012-Q5

The USC (Universal Social Charge) is calculated on gross income. The rates of the ISC are:

• 2% on the first €10,036 of gross annual income. • 4% on the next €5,980. • 7% on the balance.

(a) Niamh earned €45,000 in 2011. Find her USC for that year.

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The table shows a selection of the tax credits available in Ireland in 2011 Individual’s Tax Credits Tax Credits

2011 Single Person €16.50 Married or Civil Partner €3,300 Widowed or Surviving Civil Partner €2,190 Home Carer €810 PAYE €1,650 One Parent Family €1,650

(b) Niamh is a single person who is a PAYE worker. Calculate her total tax credits for 2011.

(c) The standard rate of tax is 20% and the higher rate is 41%. The Standard Rate Cut-off Point for a

single person is €32,800. Calculate Niamh’s tax bill for 2011.

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(d) Calculate Niamh’s net pay for the year, after tax and USC are paid.

Q2

Each week David is paid €14 per hour for the first 35 hours worked and €21 per hour for any hours worked after that.

(i) Last week he worked 38 hours. Calculate his gross pay for last week.

The standard rate of income tax is 20% and the higher rate is 41%. The standard rate cut-off point is

€230 and he has a tax credit of €62 per week.

(ii) Calculate David’s take-home pay after tax has been deducted.

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(iii) What percentage of his gross pay is his take-home pay?

Give your answer correct to the nearest whole number.

2011-Q2(b)

Aoife is single and earned €40,000 last year. Aoife’s tax credits are listed below.

Single Person Tax Credit €1,830 PAYE Tax Credit €1,830 Rent Allowance Tax Credit €400 Trade Union Payment Tax Credit €70

(i) Calculate Aoife’s total tax credits.

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The standard rate cut-off point for a single person was €36400.

The standard rate of income tax was 20% and the higher rate was 41%.

(ii) Calculate the tax paid by Aoife on her income.

Aoife also had to pay a 2% income levy on her gross income.

(iii) Calculate Aoife’s net income after all deductions had been made.

2010-Q1

(i) The standard rate of income tax is 20% and the higher rate is 41%. The standard rate cut-off point is €36,500. Aisling has a gross income of €47,500 and total tax credits of €1830. Calculate Aisling’s net income.

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(ii) The following year Aisling’s gross income increases.

The tax rates, cut-off point and tax credits remain unchanged. Her net tax income amounts to €15,105. What is her new gross income?

2006-Q1

Jill has a gross income of €50,000.

Her total income tax payable amounts to €10,460.

The standard rate cut off point is €32,000.

The standard rate of tax is 20% and the higher rate is 42%.

What are Jill’s tax credits for the year?

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2004-Q1

John has a gross income per fortnight of €1750.

The standard rate of income tax is 20% and the higher rate is 42%. He has tax credits of €105 per fortnight and his standard rate cut-off point is €1295 per fortnight.

After tax is paid, what is John’s net income per fortnight?

2005-Q2

(i) The standard rate of income tax is 20% and the higher rate is 42%. Sheila has tax credits of €2700 for the year and a standard rate cut-off point of €22,000. Sheila has a gross income of €45,000 for the year. Calculate the total tax payable by Sheila for the year.

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(ii) Tony pays tax at the same rate as Sheila. Tony has tax credits of €2900 for the year and has the same standard rate cut-off point as Sheila. His total tax payable amounts to €13,680 for the year. Calculate Tony’s gross income for the year.

2003-Q1

The standard rate of income tax is 20% and the higher rate is 42%. Fiona has tax credits of €1493 for the year and a standard rate cut-off point of €30 000.

She has a gross income of €31 650 for the year.

(i) After tax is paid, what is Fiona’s income for the year?

(ii) What would Fiona’s gross income for the year need to be in order for her to have an

after-tax income of €29 379?

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Interest/Compound Interest

Compound Interest

Very often when a sum of money earns interest, this interest is added to the principal to form a new principal.

This new principal earns interest in the next year and so on.

This is called compound interest

Formula

Use the Formula from your log tables F=P(1+i)t

When:

- The time is greater than 3 years - The rate of interest does not change - There is no changes to the amount

PIA

Use PIA method, when

- The time is 3 years or less - The amount changes through withdrawals etc. - The interest rate changes.

SEC 2014-Q4

Dermot has €5,000 and would like to invest it for two years. A special savings account is offering a rate of 3% for the first year and a higher rate for the second year, if the money is retained in the account. Tax of 33% will be deducted each year from the interest earned.

(i) How much will the investment be worth at the end of one year, after tax is deducted?

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(ii) Dermot calculates that, after tax has been deducted, his investment will be worth about €5,268 at the end of the second year. Calculate the rate of interest for the second year.

2012 -Q1

(i) €900 is invested for two years at 3% per annum compound interest. Find the value of the investment at the end of the second year.

(ii) John has a gross weekly wage of €600. After tax his net weekly wage is €554. Calculate his tax credits if he is taxed at the standard rate of 20%.

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Q2.

What sum of money invested at 2% per annum compound interest would produce interest of €306.04 after three years?

2006-Q1

A sum of money was invested at compound interest for two years.

The interest rate for each year was 5%. After the two years the sum amounted to €5512.50.

Calculate the original sum of money invested.

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2003-Q1

€6000 was invested at compound interest. The rate for the first year was 4% per annum.

(i) Calculate the amount of the investment at the end of the first year.

(ii) At the end of the second year the investment amounted to €6520.80. Calculate the rate per annum for the second year.

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2013-Q1

Ciara invested €30000 for three years at 3% per annum compound interest.

(i) Calculate the amount of the investment at the end of two years.

At the end of two years a sum of money was withdrawn. The money which remained amounted to €12,181.81 at the end of the third year.

(ii) Calculate the sum of money withdrawn at the end of two years.

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Q2.

€2000 was invested at r% for 2 years compound interest.

A tax of 25% was deducted each year from the interest gained.

At the end of the first year the investment amounted to €2030, after tax was deducted.

(i) Calculate the rate of interest r%

(ii) Find the amount of the investment at the end of 2 years, after tax has been deducted.

2008-Q1

(i) Walter borrows €5000 for three years at 4% per annum compound interest. He repays €1800 at the end of each of the first two years. How much must he repay at the end of the third year to clear his loan?

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(ii) Walter wishes to pay off his loan in equal instalments at the end of the first and second year. The rate remains at 4% per annum compound interest. How much would he need to repay, at the end of each year, to clear his loan after two years? Give your answer correct to the nearest cent.